Logic Journal of the IGPL 27 (5):659-670 (2019)

We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property is an $\mathsf{E}$-formula. Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic.
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DOI 10.1093/jigpal/jzy075
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Logic and Structure.Melvin Fitting - 1986 - Journal of Symbolic Logic 51 (3):826-827.
Intuitionistic Validity in T-Normal Kripke Structures.Samuel R. Buss - 1993 - Annals of Pure and Applied Logic 59 (3):159-173.

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